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GEOMETRIC ASPECTS OF THE REPRESENTATION THEORY
OF FINITE DIMENSIONAL ALGEBRAS
Birge Huisgen-Zimmermann (University of California at Santa Barbara)
ABSTRACT: This lecture is designed to be understood by a general mathematical audience, including graduate students with some background in basic algebraic concepts (such as would be covered in an introductory graduate course in algebra, plus low-level familiarity with the concept of an affine/projective algebraic variety).
We will start by discussing the ultimate classification goal in the study of a finite dimensional (associative) algebra $A$ over an algebraically closed field, illustrate this goal with some first examples, and introduce the concepts of finite, tame, and wild representation type. In particular, this will lead us to representations of algebras given by quiver and relations, which are located at the heart of this sequence of talks (we will introduce them here, no prior familiarity is required).
ABSTRACT: The only additional prerequisite required for this talk is the material of the overview lecture. Theory concerning algebraic group actions will be explicitly quoted as needed.
This part addresses arbitrary algebras $A$ over an algebraically closed field. We will reinforce the concept of a degeneration of a finite dimensional $A$-module with initial examples. Then we will discuss the projective varieties introduced at the end of the first lecture and exploit them to advance the degeneration theory of $A$. No proofs of the theorems will be included in the lecture (but I will be available to present proofs outside the lectures to anybody who is interested). Instead, we will emphasize concrete interpretations of the theory. In particular, we will demonstrate its force in applying it to concrete examples (these can be presented graphically, without a big buildup of technical and notational ballast, with the aid of a diagrammatic method which, in essence, was introduced by Alperin).
ABSTRACT: We explain -- first in an intuitive fashion, then more formally -- what it means for a collection of finite dimensional representations to be classifiable by way of a fine or coarse moduli space (in the sense of Mumford). Then we will answer the following question, which allows to illustrate the techniques we employ in the most transparent format: When do the representations with fixed dimension and fixed squarefree top permit such a fine or coarse classification? (Given a finite dimensional algebra $A$ with Jacobson radical $J$, the top of a left $A$-module $M$ is the semisimple module $M/JM$; that it be squarefree means that no simple summand occurs with a multiplicity larger than 1). In case the answer is positive, we describe the corresponding moduli spaces, sketch (in non-technical terms) how they can be accessed by way of quiver and relations of $A$, and again give examples. (An alternate approach to guaranteeing and constructing moduli spaces for quiver representations, due to King, will be touched on the side.)
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